Math 401 - 01 Previous Homework (Fall 2018)
Problem Set 10 (complete) Due: 11/06/2018. Board presentation: 11/20/2018
Let G be a group, and H,K≤G.
Prove that if HK=KH, then HK≤G.
Prove that if H≤NG(K), then HK≤G.
Let G be a group, H≤G, and C={gHg−1|g∈G} the set of all conjugates of H in G. Prove that: |C|=[G:NG(H)].
Let G be a group of order 120. What are the possible values of n2, n3, and n5, i.e. the number of Sylow 2-subgroups, the number of Sylow 3-subgroups and the number of Sylow 5-subgroups?
How many groups of order 6727 are there? Describe them. Justify your answers. Show all your work.
Problem Set 09 (complete) Due: 10/29/2018. Board presentation: 11/02/2018
Prove that, up to isomorphism, the direct product operation is commutative and associative.
Give an example of a group G with two subgroups H and K such that HK=G, H∩K=1, K⊴G, but G is not isomorphic to the direct product H⊕K.
Let G be a group, and H,N≤G. Prove that:
If N⊴G, then HN≤G.
If both H,N⊴G, then HN⊴G.
Make a list of all abelian groups of order 2736. Express each of them using the “elementary divisors” form and the “invariant factors” form.
Problem Set 08 (complete) Due: 10/22/2018. Problem 4 may be resubmitted by 10/24/2018. Board presentation 10/29/2018
Let n∈N and H≤G such that H is the only subgroup of G of order n. Show that H⊴G. (Do not assume that G is finite)
Let n∈N and H≤G such that H is the only subgroup of G of index n. Show that H⊴G. (Do not assume that G is finite)
Combine the previous problem with problem 3 in Problem Set 6.
Let p,q be primes such that p<q and p∤(q−1). Prove that, up to isomorphism, there is only one group of order pq. (Hint: Use example 17, page 203, as a guide. No use this example, you may use the extra assumption that (p−1)∤(q−1), or equivalently that (p−1)∤(pq−1).)
Problem Set 07 (complete) Due: 10/15/2018. Board presentation 10/29/2018
Prove Thm. 6.2.3, Thm. 6.3.2, Thm. 10.2.3. Combine all three proofs into one.
Chapter 10, problems 8, 10.
Problem Set 06 (complete) Due: 10/08/2018. Board presentation 10/10/2018
Chapter 7, problem 8.
Chapter 7, problem 22.
Let G be a finite group, and p the smallest prime divisor of |G|. If p2∤|G|, then G has at most one subgroup of index p. (Hint:Look at Example 6 on page 144)
Chapter 7, problem 12. Generalize.
Chapter 7, problem 48.
Problem Set 05 (complete) Due: 09/24/2018. Board presentation 09/28/2018
Chapter 5, problems 6, 8. For all of them find the order and the parity.
Chapter 5, problem 10. What is the largest order of an element of S8. Explain.
Chapter 5, problems 23, 24.
Chapter 5, problem 48.
Chapter 5, problem 50. Is D5 a subgroup of A5? Explain.
Problem Set 04 (complete) Due: 09/17/2018. Board presentation: 09/24/2018
Chapter 4, problem 74.
Chapter 5, problem 2.a.
Chapter 5, problem 4.
Consider α∈S8 given in disjoint cycle form by α=(1 4 5)(3 7). Write α in array form.
Problem Set 03 (complete) Due: 09/12/2018. Board presentation: 09/17/2018
Let G=⟨a⟩ be an infinite cyclic group, and k1,k2∈Z. Prove that ⟨ak1⟩≤⟨ak2⟩ iff k2∣k1.
Let G=⟨a⟩ be a cyclic group of order 60.
How many subgroups does G have?
Which of them are cyclic?
List a generator for each of the cyclic subgroups of G.
Draw the subgroup lattice of G.
Prove that a finite group of prime order must be cyclic.
Chap. 4, problem 38, 62.
Chap. 4, problem 50.
Problem Set 02 (complete) Due: 09/04/2018. Board presentation: 09/12/2018
Chap. 3, problems 4, 13, 20, 64
Chap. 3, problems 6, 50
Let G be a group in which every non-identity element has order 2. Prove that G must be Abelian.
Chap. 3, problem 34. what can you say about the union of two subgroups?
Problem Set 01 (complete) Due: 08/27/2018. Board presentation: 08/31/2018
Page 38, prob. 18. What happens if you replace each H with an A?
Page 39, prob. 22. Explain.
Page 54, prob. 4.
Page 56, prob. 22. Compare with problem 13 on page 38.
Additional problems to look at: 1.13 p.38, 1.14 p.38, 2.5 p.54,
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